In mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyperbolic manifolds are an important class of complex manifolds, defined by the property that the Kobayashi pseudometric is a metric. Kobayashi hyperbolicity of a complex manifold X implies that every holomorphic map from the complex line C to X is constant.
Definition The origins of the concept lie in Schwarz's lemma in complex analysis. Namely, if f is a holomorphic function on the open unit disc D in the complex numbers C such that f(0) = 0 and |f(z)| < 1 for all z in D, then the derivative f '(0) has absolute value at most 1. More generally, for any holomorphic map f from D to itself (not necessarily sending 0 to 0), there is a more complicated upper bound for the derivative of f at any point of D. However, the bound has a simple formulation in terms of the Poincaré metric, which is a complete Riemannian metric on D with curvature −1 (isometric to the hyperbolic plane). Namely: every holomorphic map from D to itself is distance-decreasing with respect to the Poincaré metric on D. This is the beginning of a strong connection between complex analysis and the geometry of negative curvature. For any complex space X (for example a complex manifold), the Kobayashi pseudometric dX is defined as the largest pseudometric on X such that
d X ( f ( x ) , f ( y ) ) ≤ ρ ( x , y ) {\displaystyle d_{X}(f(x),f(y))\leq \rho (x,y)} , for all holomorphic maps f from the unit disc D to X, where ρ ( x , y ) {\displaystyle \rho (x,y)} denotes distance in the Poincaré metric on D. In a sense, this formula generalizes Schwarz's lemma to all complex spaces; but it may be vacuous in the sense that the Kobayashi pseudometric dX may be identically zero. For example, it is identically zero when X is the complex line C. (This occurs because C contains arbitrarily big discs, the images of the holomorphic maps fa: D → C given by f(z) = az for arbitrarily big positive numbers a.) A complex space X is said to be Kobayashi hyperbolic if the Kobayashi pseudometric dX is a metric, meaning that dX(x,y) > 0 for all x ≠ y in X. Informally, this means that there is a genuine bound on the size of discs mapping holomorphically into X. In these terms, Schwarz's lemma says that the unit disc D is Kobayashi hyperbolic, and more precisely that the Kobayashi metric on D is exactly the Poincaré metric. The theory becomes more interesting as more examples of Kobayashi hyperbolic manifolds are found. (For a Kobayashi hyperbolic manifold X, the Kobayashi metric is a metric intrinsically determined by the complex structure of X; it is not at all clear that this should ever happen. A real manifold of positive dimension never has an intrinsic metric in this sense, because its diffeomorphism group is too big to allow that.)
Examples Every holomorphic map f: X → Y of complex spaces is distance-decreasing with respect to the Kobayashi pseudometrics of X and Y. It follows that if two points p and q in a complex space Y can be connected by a chain of holomorphic maps C → Y, then dY(p,q) = 0, using that dC is identically zero. This gives many examples of complex manifolds for which the Kobayashi pseudometric is identically zero: the complex projective line CP1 or more generally complex projective space CPn, C−{0} (using the exponential function C → C−{0}), an elliptic curve, or more generally a compact complex torus. Kobayashi hyperbolicity is preserved under passage to open subsets or to closed complex subspaces. It follows, for example, that any bounded domain in Cn is hyperbolic. A complex space is Kobayashi hyperbolic if and only if its universal covering space is Kobayashi hyperbolic. This gives many examples of hyperbolic complex curves, since the uniformization theorem shows that most complex curves (also called Riemann surfaces) have universal cover isomorphic to the disc D. In particular, every compact complex curve of genus at least 2 is hyperbolic, as is the complement of 2 or more points in C.
Basic results For a Kobayashi hyperbolic space X, every holomorphic map C → X is constant, by the distance-decreasing property of the Kobayashi pseudometric. This is often the most important consequence of hyperbolicity. For example, the fact that C minus 2 points is hyperbolic implies Picard's theorem that the image of any nonconstant entire function C → C misses at most one point of C. Nevanlinna theory is a more quantitative descendant of Picard's theorem. Brody's theorem says that a compact complex space X is Kobayashi hyperbolic if and only if every holomorphic map C → X is constant. An application is that hyperbolicity is an open condition (in the Euclidean topology) for families of compact complex spaces. Mark Green used Brody's argument to characterize hyperbolicity for closed complex subspaces X of a compact complex torus: X is hyperbolic if and only if it contains no translate of a positive-dimensional subtorus. If a complex manifold X has a Hermitian metric with holomorphic sectional curvature bounded above by a negative constant, then X is Kobayashi hyperbolic. In dimension 1, this is called the Ahlfors–Schwarz lemma.
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